Let F(x) be a polynomial with integer coefficients.
There are two distinct points on the graph of F, say P and Q, with integer coordinates.
If the length of PQ is an integer, then will PQ always be parallel to the x-axis?
If so, prove it.
If not, provide an example.
(In reply to
re: Seems unlikely by Jer)
I think you have it there - that for a non-horizontal line we need a Pythagorean triple with "1", which doesn't happen.
Let's start from the beginning.
We can translate the curve so one of the two points is the origin. Since the points are lattice points the translation will not affect the property that the coefficients are integers.
And both slope and distances are preserved over translations so the slope and distance between P and Q will be the same as the translated version of P and Q.
Let's call the translated curve F'(x)
So now (0,0) is a point on F'(x). Then F'(0)=0, which means the constant term is zero, and subsequently, x is a factor if F'(x). Then let G(x) be the rest of F'(x), that is F'(x) = x*G(x).
Now let (b,c) be the other point. Then c = b*G(b). b, c, and all the coefficients of G(x) are integers. Because integers are closed under addition and multiplication then G(b) is also an integer.
This means that b*G(b) is a factorization of c, which means c is a multiple of b.
But then the distance from (0,0) to (b,c) is an integer, call that d. This means by the Pythagorean theorem (or distance formula) we have b^2+c^2=d^2. Now substitute c = b*G(b) and get c^2*(1 + G(b)^2) = d^2.
Everything is supposed to be integers, so then that means c must divide d. Let d = c*n. Then the Pythagorean equation reduces to 1+G(b)^2 = n^2.
Over the integers the only way to satisfy this relation is when G(b)=0. But then c=b*G(b) reduces to c=0. Then the two points are (0,0) and (b,0) which describes a horizontal line coincident to the x-axis.
But then undoing the transformation applied at the beginning will still leave us with a horizontal line PQ parallel to the x-axis. So then we conclude that for the given system in the problem if the length of PQ is an integer, then PQ will always be parallel to the x-axis.
re(2): Seems unlikely - Solution
